Secondary Mathematics: Strategic Thinking

This is a secondary elective mathematics course created by VL Gaffney for independent study students of higher maths. Strategic thinking represents a block of analytical learning that is essential for problem solving and developing unique approaches to the challenges of every day life, even those unrelated to mathematic reasoning. This course and its content were made possible by EdX, champions of freely accessible education, and the inspiring natures of Dr. Michael Starbird (University of Texas) and Mr. Jordan Marks (formerly of Hendrickson HS.)

Recommended text: The Heart of Mathematics: an Invitation to Effective Thinking

Five Strategies for Solution Finding
1. Understand what is really being asked of you
2. Be willing to make mistakes, accept mistakes as part of the process of problem solving
3. Raise questions, as many as it takes
4. Follow the flow of ideas, but don’t get ahead of yourself
5. Accept change as the only universal constant, embrace that you may grow from what you learn

Course Activities to Implement the Given Strategies
A. You have nine visually identical objects, and two single-use balance scales (each can only be used once, for one balancing.) One of the nine objects weighs almost imperceptibly more than the other eight, so you must use the scales to find the slightly heavier one. Can this be done? How, or why not?
B. Five couples, ten people, are attending a party, including the host couple. At some point during the event, one of the hosts asks the other nine attendees how many hands they’ve shaken that evening. Each of the other nine answer differently. One says they’ve shaken no hands. One says they’ve shaken one other’s hand. One says they’ve shaken two, one says they’ve shaken three, one says they’ve shaken four, on up in this way to one saying they’ve shaken eight hands. How many hands did the partner of the host shake, if no one shook the hand of their partner, or their own hand (for obvious reasons)?
C. Two groups of six travelers, three in group A and three in B, are traversing a terrain and come to a river they must cross. It is too wide and deep to swim. There is a rowboat that seats just two. The travelers know that at no time can the number of members of group A outnumber the members of group B on either side of the river. What is the sequence of river crossings to move all six travelers across the river, never breaking the condition that the groups on either side of the river must be at least equal if not B-dominant at all times?

D. Write a reflective essay on how the 5 Strategies are implemented, detail how you implemented them to come to the solutions of the activity puzzles.
E. Write a puzzle of your own, and explain how it is solved using any or all of the 5 Strategies
F. Create a tutorial video on how to solve any or all of the activity puzzles, implementing the 5 Strategies.
G. Write a reflective essay on how the 5 Strategies relate to every day life, outside of math and puzzles. How did working on these puzzles frustrate or challenge you? Did you learn anything, other than the simple solutions to the puzzles?

More activities:
Practice on the Hanoi Towers simulator, using no fewer than five rings. Larger rings cannot be placed on top of smaller rings, and only one ring can be moved at a time.

DO NOT CONTINUE BELOW TO SOLUTIONS UNTIL YOU’VE COMPLETED THE PUZZLE ACTIVITIES INDEPENDENTLY.

Puzzle Activity 1: Yes. Sort the nine objects into three groups of three objects. Weigh two of the groups on the first single-use scale. If one group weighs more than the other, the heavier object is in that group. If the groups weigh the same, the heaver object is in the third yet-unweighed group. Next, weigh two of the three remaining objects. If one weighs more, you’ve identified the one out of nine. If the two weigh the same, the final remaining unweighed is the one of nine that weighs more than the other eight.
Puzzle Activity 2: The asking host’s spouse/partner shook four hands. If you simplified the problem to fewer couples, you’d discover that the person who shook 8 hands must be partners with the person who shook no (0) hands. If you then created a chart to visualize each of the partner pairs (A/B, C/D, E/F, G/H, I/J) and how many hands each shook, you’d find that two attendees must have shaken four hands each. Since only one person responds that they shook four hands, but you know that two of the ten shook four hands, the host must be one and their partner must be the other.
Puzzle Activity 3: It takes 21 steps to move all of the travelers safely across the river, including a step that seems counter-productive. In Step 10, there are four travelers across the river, but two must turn back to maintain the rule, resulting in four travelers on the ‘wrong’ side of the river in Step 12. Yet, it’s a necessary part of the steps and brings you to the solution quickly. This puzzle is sometimes presented as “Pirates and Captains,” or “Wolves and Sheep,” where the A group is predatory or dangerous to the B group, which is why A can’t outnumber B.

Hanoi Towers: With five rings, it takes a minimum of 31 moves to ‘win’ the game. With 8, 255 moves. This is not a puzzle to attempt on paper.